cos 2θ

Formula

$\cos 2\theta = \cos^2 \theta -\sin^2 \theta$

It is read as cos two theta is equal to cos squared theta minus sin squared theta.

Proof

sine two theta right angled triangle
Step: 1

According to $\Delta UOS$

$2\theta$ is the angle of the right angled triangle $UOS$ and express cosine in terms of its ratio form.

$\cos 2 \theta = \dfrac{OU}{OS}$

The length of the side $\overline{OU}$ can be expressed as the subtraction of the length of the side $\overline{UP}$ from the side $\overline{OP}$.

$\implies \cos 2 \theta = \dfrac{OP -UP}{OS}$

$\implies \cos 2 \theta = \dfrac{OP}{OS} -\dfrac{UP}{OS}$

The length of the side $\overline{UP}$ is exactly equal to the length of the side $\overline{WR}$, it means $UP = WR$. So, the length of the side $\overline{UP}$ can be replaced by the length of the side $\overline{WR}$.

$\implies \cos 2 \theta = \dfrac{OP}{OS} -\dfrac{WR}{OS}$

Step: 2

According to $\Delta POR$

right angled triangle por

Express cosine in its ratio form according to right angled triangle $POR$.

$\cos \theta = \dfrac{OP}{OR}$

$\implies OP = OR \cos \theta$

Now, replace the length of the side $\overline{OP}$ by its new value to transform the $\cos 2\theta$ equation.

$\implies \cos 2 \theta = \dfrac{OR \cos \theta}{OS} -\dfrac{WR}{OS}$

$\implies \cos 2 \theta = \dfrac{OR}{OS} \times \cos \theta -\dfrac{WR}{OS}$

Step: 3

According to $\Delta WSR$

right angled triangle wsr

Consider right angled triangle $WSR$ and express sine in its ratio form to transform the length of the side $\overline{WR}$ in alternative form.

$\sin \theta = \dfrac{WR}{SR}$

$\implies WR = SR \sin \theta$

Replace the length of the side $\overline{WR}$ by its new value in $\cos 2\theta$ equation.

$\implies \cos 2\theta = \dfrac{OR}{OS} \times \cos \theta -\dfrac{SR \sin \theta}{OS}$

$\implies \cos 2\theta = \dfrac{OR}{OS} \times \cos \theta -\dfrac{SR}{OS} \times \sin \theta$

Step: 4

According to $\Delta ROS$

right angled triangle ros

Consider the right angled triangle $ROS$ In order to transform the $\cos 2\theta$ expansion purely in terms of trigonometric functions sine and cosine.

$\cos \theta = \dfrac{OR}{OS}$

$\sin \theta = \dfrac{SR}{OS}$

Now, substitute the ratios by its respective trigonometric functions in the $\cos 2\theta$ equation.

$\implies \cos 2 \theta = \cos \theta \times \cos \theta -\sin \theta \times \sin \theta$

$\therefore \,\,\,\,\,\,\, \cos 2 \theta = \cos^2 \theta -\sin^2 \theta$

In this way, cosine of double angle is expressed in terms of cosine and sine of the angle.

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